Problem: Simplify the following expression and state the condition under which the simplification is valid. $p = \dfrac{4n^2 - 8n - 32}{8n^3 - 40n^2 + 32n}$
Answer: First factor out the greatest common factors in the numerator and in the denominator. $ p = \dfrac {4(n^2 - 2n - 8)} {8n(n^2 - 5n + 4)} $ $ p = \dfrac{4}{8n} \cdot \dfrac{n^2 - 2n - 8}{n^2 - 5n + 4} $ Simplify: $ p = \dfrac{1}{2n} \cdot \dfrac{n^2 - 2n - 8}{n^2 - 5n + 4}$ Next factor the numerator and denominator. $ p = \dfrac{1}{2n} \cdot \dfrac{(n - 4)(n + 2)}{(n - 4)(n - 1)}$ Assuming $n \neq 4$ , we can cancel the $n - 4$ $ p = \dfrac{1}{2n} \cdot \dfrac{n + 2}{n - 1}$ Therefore: $ p = \dfrac{ n + 2 }{ 2n(n - 1)}$, $n \neq 4$